is an ultrafilter of events on \({Z}\) ; see for instance [1]} and [2]}.
If the ultrafilter \({U}_{{\underline{P}_{}},{Z_{{}}}}\)  is fixed, meaning that \(\bigcap {U}_{{\underline{P}_{}},{Z_{{}}}}={z_o}\)  for some \(z_o\in {Z}\) , then \(P_{Z_{{}}}(h)=h(z_o)\) , so all probability mass of the precise distribution \(P_{Z_{{}}}\)  is concentrated in \(z_o\) .
The only other possibility is that the ultrafilter \({U}_{{\underline{P}_{}},{Z_{{}}}}\)  is free, meaning that \(\bigcap {U}_{{\underline{P}_{}},{Z_{{}}}}=\emptyset \) , and then typically all probability mass will lie infinitesimally close to some \(z_o\)  in \({Z}\) , or to some `point on the boundary' of \({Z}\) .
In both cases, this represents a model for our subject's certainty that \(Z_{{}}\)  assumes a fixed value; see also the extensive discussion in Section 5.5.5 of [1]}.
